Structure of the quasihyperbolic stochasticity in an inertial self-excited oscillator

Abstract

The foregoing investigations of the dynamics of the system (1), the results of which are largely similar to those obtained in our numerical analysis of Rossler's system [13] and to many other published data [1, 3–5, 11, 14, 15], indicate that quasihyperbolic attractors with a small fractional dimension in the domain above the critical state are typified by soft and hard bifurcations of the periodic trajectories. A bifurcation generating a stable two-dimensional torus from a cycle in nonautonomous three-dimensional systems in which the degree of contraction of the phase volume depends on the coordinates is clearly not realized [15], and for this effect to exist it is necessary to increase the dimension of the phase space to N=4 by introducing, for example, an external control signal. An analysis of systems with a forward Andronov-Hopf bifurcation shows that the typical mechanism of the evolution of stochasticity here is the mechanism of Feigenbaum doublings, against the background of which the mixing properties of hard bifurcations occur in the supercritical zone.